Abstract

There are two parts in this dissertation: the chromatic equivalence classes and
the chromatic defining numbers of graphs.
In the first part the chromaticity of the family of generalized polygon trees with
intercourse number two, denoted by Cr (a, b; c, d), is studied. It is known that
Cr( a, b; c, d) is a chromatic equivalence class if min {a, b, c, d} ≥ r+3. We consider
Cr( a, b; c, d) when min{ a, b, c, d} ≤ r + 2. The necessary and sufficient conditions
for Cr(a, b; c, d) with min {a, b, c, d} ≤ r + 2 to be a chromatic equivalence class
are given. Thus, the chromaticity of Cr (a, b; c, d) is completely characterized.
In the second part the defining numbers of regular graphs are studied. Let
d(n, r, X = k) be the smallest value of defining numbers of all r-regular graphs
of order n and the chromatic number equals to k. It is proved that for a given
integer k and each r ≥ 2(k - 1) and n ≥ 2k, d(n, r, X = k) = k - 1. Next,
a new lower bound for the defining numbers of r-regular k-chromatic graphs
with k < r < 2( k - 1) is found. Finally, the value of d( n , r, X = k) when
k < r < 2(k - 1) for certain values of n and r is determined.